Let $X=A\cup B$ be a closed bundle-gluing cover in the topological or smooth category, write $C=A\cap B$, and let $(E_A,E_B,\varphi)$ be a gluing datum for rank $k$ real vector bundles in that category. Assume that the collars of $C$ in $A$ and $B$ admit bundle trivializations compatible with the product collar coordinates, that the induced neighbourhood of $C$ in $X$ is obtained by gluing these two collars to $C\times(-\varepsilon,\varepsilon)$ for some $\varepsilon>0$, and that in these trivializations the boundary identification $\varphi$ is represented by a continuous, respectively smooth, map $C\to GL(k,\mathbb R)$. Define $E=(E_A\sqcup E_B)/\sim$, where $e_A\in E_A|_{C}$ is identified with $\varphi(e_A)\in E_B|_{C}$. Then $E\to X$ is a rank $k$ real vector bundle whose restrictions to $A$ and $B$ recover $E_A$ and $E_B$ up to the prescribed identification on $C$.